Introduction to Nilpotent Groups (original) (raw)
Related papers
The Theory of Nilpotent Groups
The Theory of Nilpotent Groups, 2017
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On the Nilpotency of a Pair of Groups
Southeast Asian Bulletin of …, 2012
This paper is devoted to suggest that the extensive theory of nilpotency, upper and lower central series of groups could be extended in an interesting and useful way to a theory for pairs of groups. Also this yields some information on nilpotent groups.
Nilpotency: A Characterization Of The Finite p-Groups
Journal of Mathematics , 2017
Abstract As parts of the characterizations of the finite p-groups is the fact that every finite p-group is NILPOTENT. Hence, there exists a derived series (Lower Central) which terminates at e after a finite number of steps. Suppose that G is a p-group of class at least m ≥ 3. Then L m-1G is abelian and hence G possesses a characteristic abelian subgroup which is not contained in Z(G). If L 3(G) = 1 such that pm is the highest order of an element of G/L2 (G) (where G is any p-group) then no element of L2(G) has an order higher than pm. [1]
On a Class of Generalized Nilpotent Groups
Journal of Algebra, 2002
We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.
Rocky Mountain Journal of Mathematics, 1973
In this article a chain condition for groups, called the bounded chain condition, is studied; this chain condition includes the ascending and descending chain conditions as special cases. It is shown that every locally radical group which satisfies the bounded chain condition on subgroups must satisfy the ascending or descending chain condition on subgroups, i.e., such groups are Artinian or Noetherian. The same conclusion holds for nilpotent groups which satisfy the bounded chain condition on abelian subgroups.
Glasgow Mathematical Journal
Abstarct Let γ n = [x1,…,x n ] be the nth lower central word. Denote by X n the set of γ n -values in a group G and suppose that there is a number m such that ∣gXn∣lem|{g^{{X_n}}}| \le m∣gXn∣lem for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.
Groups that are Pairwise Nilpotent
Communications in Algebra, 2008
In this paper we study groups generated by a set X with the property that every two elements in X generate a nilpotent subgroup.
On various concepts of nilpotence for expansions of groups
Publicationes Mathematicae Debrecen, 2013
The group theoretic concept of nilpotence has been generalized in various ways to arbitrary universal algebras. We establish a relation between two such generalizations for expansions of groups.
Israel Journal of Mathematics, 2008
We define a notion of roundness for finite groups. Roughly speaking, a group is round if one can order its elements in a cycle in such a way that some natural summation operators map this cycle into new cycles containing all the elements of the group. Our main result is that this combinatorial property is equivalent to nilpotence.